Determine the values of trigonometric functions using the unit circle
Answer 1
Isabella Walker
To find the exact values of the trigonometric functions for the angle $ \theta = \frac{5\pi}{4} $ using the unit circle, follow these steps:
1. Locate the angle $ \theta = \frac{5\pi}{4} $ on the unit circle. This angle corresponds to $ 225^{\circ} $, or $ 45^{\circ} $ in the third quadrant.
2. In the third quadrant, both the sine and cosine values are negative. The reference angle is $ 45^{\circ} $.
3. The coordinates for $ 45^{\circ} $ are $ \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) $, so for $ 225^{\circ} $ these coordinates are $ \left( -\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2} \right) $.
4. Therefore, $ \sin\left( \frac{5\pi}{4} \right) = -\frac{\sqrt{2}}{2} $ and $ \cos\left( \frac{5\pi}{4} \right) = -\frac{\sqrt{2}}{2} $.
5. The tangent function is $ \tan\left( \frac{5\pi}{4} \right) = \frac{\sin\left( \frac{5\pi}{4} \right)}{\cos\left( \frac{5\pi}{4} \right)} = \frac{-\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = 1 $.
Answer 2
Mia Harris
To determine $ sinleft( frac{5pi}{4}
ight) $, $ cosleft( frac{5pi}{4}
ight) $, and $ anleft( frac{5pi}{4}
ight) $, follow these steps:
1. Convert $ frac{5pi}{4} $ radians to degrees: $ frac{5pi}{4} imes frac{180}{pi} = 225^{circ} $.
2. The angle $ 225^{circ} $ is in the third quadrant where sine and cosine are negative.
3. Reference angle is $ 45^{circ} $: coordinates $ left( frac{sqrt{2}}{2}, frac{sqrt{2}}{2}
ight) $ become $ left( -frac{sqrt{2}}{2}, -frac{sqrt{2}}{2}
ight) $.
4. Thus, $ sinleft( frac{5pi}{4}
ight) = -frac{sqrt{2}}{2} $, $ cosleft( frac{5pi}{4}
ight) = -frac{sqrt{2}}{2} $, and $ anleft( frac{5pi}{4}
ight) = 1 $.
Answer 3
Henry Green
For $ heta = frac{5pi}{4} $:
$ sinleft( frac{5pi}{4}
ight) = -frac{sqrt{2}}{2} $
$ cosleft( frac{5pi}{4}
ight) = -frac{sqrt{2}}{2} $
$ anleft( frac{5pi}{4}
ight) = 1 $
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